On the fourth-order Leray-Lions problem with indefinite weight and nonstandard growth conditions

We prove the existence of at least three weak solutions for the fourth-order problem with indefinite weight involving the Leray-Lions operator with nonstandard growth conditions. The proof of our main result uses variational methods and the critical theorem of Bonanno and Marano (Appl. Anal. 89 (2010), 1-10).


Introduction
In this paper, we shall show the existence of three weak solutions for the following interesting problem ∆ (a(x, ∆u)) = λV (x)|u| q(x)−2 u in Ω, u = ∆u = 0 on ∂Ω, (1.1) where Ω is a bounded domain in R N (N ≥ 2) with a smooth boundary ∂Ω, λ > 0 is a parameter, V is a function in a generalized Lebesgue space L s(x) (Ω), functions p, q, s ∈ C(Ω) satisfy the inequalities for all x ∈ Ω, and ∆ (a(x, ∆u)) is the Leray-Lions operator of the fourth-order, where a is a Carathéodory function satisfying some suitable supplementary conditions. For more details about this kind of operators the reader is referred to Boureanu [4] and Leray-Lions [17] (and the references therein).
Note that the study of this type of operators is very active in several fields, e.g. in electrorheological fluids (Růžička [22]), elasticity (Zhikov [24]), stationary thermorheological viscous flows of non-Newtonian fluids (Rajagopal-Růžička [21]), image processing (Chen-Levine-Rao [5]), and mathematical description of the processes filtration of barotropic gas through a porous medium (Antontsev-Shmarev [2]). Similar problems have been studied before by various authors, see e.g. recent papers of Afrouzi-Chung-Mirzapour [1], Kefi-Rȃdulescu [14], Kong [15,16], and Chung-Ho [6]. In particular, Kefi [13] studied the following problem Under the condition in problem (1.1), he has shown that problem (1.2) has a continuous spectrum and his main argument was the Ekeland variational principal. Before introducing our main result, we define and for η > 0, h ∈ C + (Ω), we set Remark 1.1. It is easy to verify that the following holds We denote where B is the ball of radius δ centered at x. One can prove that there exists x 0 ∈ Ω such that B(x 0 , D) ⊆ Ω, where D := sup x∈Ω δ(x).
Throughout this paper, we shall need the following hypotheses: (H 1 ) a : Ω × R → R is a Carathéodory function such that a(x, 0) = 0, for a.e. x ∈ Ω.
(H 2 ) There exist c 1 > 0 and a nonnegative function α ∈ L p(x) , for a.e. x ∈ Ω and all t ∈ R.
(H 3 ) The following inequality holds (a(x, t) − a(x, s))(t − s) ≥ 0, for a.e. x ∈ Ω, and all s, t ∈ R with equality if and only if s = t.
(H 4 ) The following inequality holds where A : Ω × R → R represents the antiderivative of a, that is, is the ball of radius D centered at x 0 and v 0 is a positive constant. Remark 1.2. We note the following facts: (1) A(x, t) is a C 1 -Carathéodory function, i.e., for every t ∈ R, A(., t) : Ω → R is measurable and A(x, .) ∈ C 1 (R), for a.e. x ∈ Ω.
(2) By hypothesis (H 2 ), there exists a constant c 3 such that , for a.e. x ∈ Ω and all t ∈ R.
In the sequel, let , where Γ denotes the Euler function. Furthermore, let k > 0 be the best constant for which the inequality (2.2) below holds. The main result of this paper now reads as follows.
Theorem 1.1. Assume that hypotheses (H 1 ) − (H 5 ) are fulfilled and that there exist r > 0 and d > 0 such that and Then for every λ ∈ Λ r : Remark 1.3. If we set r = 1, then conditions of Theorem 1.1 read as follows: There exists d > 0 such that Remark 1.4. We are interested in the Leray-Lions type operators because they are quite general. Indeed, consider where p ∈ C + (Ω), p + < +∞, and choose θ ∈ L ∞ (Ω) such that there exists x ∈ Ω. One can then see that (1.6) satisfies hypotheses (H 1 ) − (H 4 ) and we arrive at the following operator Note that when θ ≡ 1, we get the well-known p(x)-biharmonic operator ∆ 2 p(.) (u), see Kefi-Rȃdulescu [14]. Moreover, we can make the choice and obtain the following operator where p and θ are as in (1.6).
In the sequel, define a(x, t) as in (1.6) with θ ≡ 1. Then problem (1.1) becomes and we obtain the following result.
Then for every problem (1.7) admits at least three weak solutions.
This paper is organized as follows: in Section 2, we give some preliminaries and necessary background results on the Sobolev spaces with variable exponents, whereas Section 3 is devoted to the proof of our main result.

Preliminaries and Background
In this section, we recall some definitions and basic properties of variable exponent Sobolev spaces. For a deeper treatment of these spaces, we refer the reader to Fan-Zhao [10], Rȃdulescu [19], and Rȃdulescu-Repovš [20], and for the other background material to Papageorgiou-Rȃdulescu-Repovš [18].
Let p ∈ C + (Ω) be such that We define the Lebesgue space with variable exponent as follows which is equipped with the so-called Luxemburg norm Variable exponent Lebesgue spaces are like classical Lebesgue spaces in many respects: they are Banach spaces and are reflexive if and only if 1 < p − ≤ q + < ∞. Moreover, the inclusion between Lebesgue spaces is generalized naturally: if q 1 , q 2 are such that p 1 (x) ≤ p 2 (x), a.e. x ∈ Ω, then there exists a continuous embedding For u ∈ L p(x) (Ω) and v ∈ L p ′ (x) (Ω), the Hölder inequality holds The modular on the space L p(x) (Ω) is the map ρ p(x) : L p(x) (Ω) → R defined by For any positive integer m, we define the Sobolev space with variable exponents as follows: where α := (α 1 , α 2 , ..., α N ) is a multi-index and Then W m,p(x) (Ω) is a separable and reflexive Banach space equipped with the norm (Ω) is the closure of C ∞ 0 (Ω) in W m,p(x) (Ω). It's wellknown that both W 2,p(x) (Ω) and W 1,p(x) 0 (Ω) are separable and reflexive Banach spaces. It follows that is also a separable and reflexive Banach space, when equipped with the norm represent a norm which is equivalent to . X on X (see El Amrouss-Ourraoui [9, Remark 2.1]). Therefore in what follows, we shall consider the normed space (X, . ) .
Proposition 2.1. (Edmunds-Rakosnik [7]) Let p and q be measurable functions such that p ∈ L ∞ (Ω), and 1 ≤ p(x)q(x) ≤ ∞, for a.e. x ∈ Ω. Let u ∈ L q(x) (Ω), u = 0. Then We recall that the critical Sobolev exponent is defined as follows: Remark 2.1. (Kefi [13]) Denote the conjugate exponent of the function . Then there exist compact and continuous embeddings X ֒→ L s ′ (x)q(x) (Ω) and X ֒→ L β(x) (Ω) and the best constant k > 0 such that In order to formulate the variational approach to problem (1.1), let us recall the definition of a weak solution for our problem.
We state the following proposition which will be needed in Section 3. [11]) If X is a reflexive Banach space, Y is a Banach space, Z ⊂ X is nonempty, closed and convex subset, and J : Z → Y is completely continuous, then J is compact.

Proposition 2.2. (Gasiński-Papageorgiou
Our main tool will be the following critical theorem Bonanno-Marano [3], which we restate in a more convenient form. Theorem 3.6]) Let X be a reflexive real Banach space and Φ : X → R a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X. Let Ψ : X → R be a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that Assume that there exist r > 0 and x ∈ X, with r < Φ(x), such that: , the functional Φ−λΨ is coercive.
Then for each λ ∈ Λ r , the functional Φ − λΨ has at least three distinct critical points in X.

Proof of the Main Result
In this section, we present the proof of Theorem 1.1. To begin, let us denote The Euler-Lagrange functional corresponding to problem (1.1) is then defined by I λ : X → R, It is clear that condition (a 0 ) in Theorem 2.1 is fulfilled, and by virtue of Proposition 2.1, Ψ is well-defined since we have for all u ∈ X, Moreover, by inequality (2.2) in Remark 2.1, one has therefore Ψ is indeed well-defined. We shall also need the following lemma.
The functional Φ is a coercive, continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional (Boureanu [4]) whose Gâteaux derivative admits a continuous inverse on X.
(ii) The functional Ψ is a a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact.
Proof. The proof splits into two parts: (i) It is clear from Lemma 2.1 and hypothesis (H 4 ) that for every u ∈ X such that u > 1, one has and thus Φ is coercive.
For the rest of the proof, we will use the same argument as in the proof of Ho-Sim [12, Lemma 3.2]. First, we shall show that Φ ′ is strictly monotone. Using (H 3 ) and integrating over Ω, we obtain for all u, v ∈ X with u = v, which means that Φ ′ is strictly monotone.
Note that the strict monotonicity of Φ ′ implies that Φ ′ is an injection. From the assertion (H 4 ) it is clear that for any u ∈ X with u > 1, one has and thus Φ ′ is coercive. Therefore it is a surjection in view of Minty-Browder Theorem for reflexive Banach space (cf. Zeidler [23]), so Φ ′ has a bounded inverse mapping (Φ ′ ) −1 : X * → X.
Let f n → f as n → +∞ in X * and set u n = ( Then the boundedness of (Φ ′ ) −1 and {f n } imply that {u n } is bounded. Without loss of generality, we can assume that there exists a subsequence, again denoted by u n , andũ such that u n ⇀ũ (weakly) in X, which implies We can now infer that n→+∞ Ω a(x, ∆u n )(∆u n − ∆ũ)dx = 0.
(ii) Next, we show that Ψ ′ (u) is compact. Let v n ⇀ v in X. Then As a consequence of Remark 2.1 and due to the compact embedding , v > |, as n → +∞. This means that Ψ ′ (u) is completely continuous. So, by Proposition 2.2, Ψ ′ is indeed compact.
Proof of Theorem 1.1. As we have observed above, the functionals Φ and Ψ satisfy the regularity assumptions of Theorem 2.1. Now, let v d ∈ X be the function defined by where |.| denotes the Euclidean norm in R N . It is then easy to see that .
Using Lemma 2.1 and the continuity of the embedding L p + (Ω) ֒→ L p(x) (Ω), we can conclude that and hence Next, from r < 1 p + 2d(N −1) D 2 p L, we get r < Φ(v d ). Now, for each u ∈ Φ −1 ((−∞, r]), due to condition (H 4 ), one has that 1 p + u p ≤ r.  In the next step, we shall prove that for each λ > 0, the energy functional Φ − λΨ is coercive. By Remark 2.1, we have Since 1 ≤ q − ≤ q + < p − , it follows that Φ(u) − λΨ(u) is coercive. Finally, due to the fact that Theorem 2.1 implies that for each λ ∈ Λ r , the functional Φ − λΨ admits at least three critical points in X which are weak solutions for problem (1.1). This completes the proof of Theorem 1.1.